I don't know if it's even stupid as history--history's a lot more contingent than we think, and there's no reason the Romans couldn't have been wiped out, changing the course of history forever.
Whatever the historical inaccuracy, it's AMAZING as a game idea. Good excuse to go back and read every Greek mythology book you've got. Mythic Odysseys of Theros is the obvious starting off point. I've seen an OSR game, Mazes & Minotaurs, that does 'what if D&D was based on ancient Greece not medieval Europe' you might also want to mine for ideas. Going a little farther afield, Conan games are usually supposed to mimic the ancient rather than medieval world and might also be good sources to steal from. It's D&D, not history.
There are many classic Greek monsters already in the game like harpies, manticores, griffins, sphinxes, giants, and centaurs, and in some editions cyclopes; you could flesh those out into complete societies with a few different cultures. There's also the singular creatures D&D turned into monster species like the Minotaur, Medusa (one of 3), Lamia, Chimera, and Hydra; you could also expand on Echidna, Scylla and Charybdis, Cerberus, Python the serpent, the Colchian dragon, the Nemean Lion, the man-eating horses of Diomedes, the Stymphalian birds, sirens,
D&D with its polytheism is in some ways a better fit for the ancient than the medieval world religiously if not technologically.
If you're nuts enough you can serve Mediterranean food instead of pizza--ancient Greeks had grapes, olives, and cheese, all of which are able to be eaten as snacks instead of the traditional chips or pretzels. They drank wine, but you may not want drunk players, especially if you have to drive home...but you could do grape juice.